CONTROL SYSTEMS
Control is used to modify the behavior of a system so it behaves in a specific desirable way over time. For
example, we may want the speed of a car on the highway to remain as close as possible to 60 miles per hour
in spite of possible hills or adverse wind; or we may want an aircraft to follow a desired altitude, heading,
and velocity profile independent of wind gusts; or we may want the temperature and pressure in a reactor
vessel in a chemical process plant to be maintained at desired levels. All these are being accomplished today
by control methods and the above are examples of what automatic control systems are designed to do,
without human intervention. Control is used whenever quantities such as speed, altitude, temperature, or
voltage must be made to behave in some desirable way over time.
This section provides an introduction to control system design methods.P.A., Z.G.
In This Section:
CHAPTER 19.1 CONTROL SYSTEM DESIGN 19.3
INTRODUCTION 19.3
ProportionalIntegralDerivative Control 19.3
The Role of Control Theory 19.4
MATHEMATICAL DESCRIPTIONS 19.4
Linear Differential Equations 19.4
State Variable Descriptions 19.5
Transfer Functions 19.7
Frequency Response 19.9
ANALYSIS OF DYNAMICAL BEHAVIOR 19.10
System Response, Modes and Stability 19.10
Response of First and Second Order Systems 19.11
Transient Response Performance Specifications for a Second Order Underdamped System 19.13
Effect of Additional Poles and Zeros 19.14
CLASSICAL CONTROL DESIGN METHODS 19.14
Design Specifications and Constraints 19.14
Control Design Strategy Overview 19.15
Evaluation of Control System 19.19
Digital Implementation 19.20
ALTERNATIVE DESIGN METHODS 19.21
Nonlinear PID 19.21
State Feedback and Observer BasedDesign 19.22
SECTION 19
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.1
The Electronics Engineers' Handbook, 5th Edition
McGrawHill, Section 19, pp. 19.119.30, 2005.
ADVANCED ANALYSIS AND DESIGN TECHNIQUES 19.26
APPENDIX: OPEN AND CLOSED LOOP STABILIZATION 19.27
REFERENCES 19.29
On the CDROM:
“A Brief Review of the Laplace Transform,” by the authors of this section, examines its use fulness in
control Functions.
19.2
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.2
19.3
CHAPTER 19.1
CONTROL SYSTEM DESIGN
Panus Antsaklis,Zhiqiany Gao
INTRODUCTION
To gain some insight into how an automatic control system operates we shall briefly examine the speed con
trol mechanism in a car.
It is perhaps instructive to consider first how a typical driver may control the car speed over uneven ter
rain. The driver, by carefully observing the speedometer, and appropriately increasing or decreasing the fuel
flow to the engine, using the gas pedal, can maintain the speed quite accurately. Higher accuracy can perhaps
be achieved by looking ahead to anticipate road inclines. An automatic speed control system, also called
cruise control, works by using the difference, or error, between the actual and desired speeds and knowledge
of the car’s response to fuel increases and decreases to calculate via some algorithm an appropriate gas pedal
position, so to drive the speed error to zero. This decision process is called a control law and it is implemented
in the controller. The system configuration is shown in Fig. 19.1.1. The car dynamics of interest are captured
in the plant. Information about the actual speed is fed back to the controller by sensors, and the control deci
sions are implemented via a device, the actuator, that changes the position of the gas pedal. The knowledge
of the car’s response to fuel increases and decreases is most often captured in a mathematical model.
Certainly in an automobile today there are many more automatic control systems such as the antilock brake
system (ABS), emission control, and tracking control. The use of feedback control preceded control theory,
outlined in the following sections, by over 2000 years. The first feedback device on record is the famous Water
Clock of Ktesibios in Alexandria, Egypt, from the third century BC.
ProportionalIntegralDerivative Control
The proportionalintegralderivative (PID) controller, defined by
(1)
is a particularly useful control approach that was invented over 80 years ago. Here K
P
, K
I
, and K
D
are controller
parameters to be selected, often by trial and error or by the use of a lookup table in industry practice. The goal,
as in the cruise control example, is to drive the error to zero in a desirable manner. All three terms Eq. (1) have
explicit physical meanings in that e is the current error, ∫e is the accumulated error, and represents the trend.
This, together with the basic understanding of the causal relationship between the control signal (u) and the
output (y), forms the basis for engineers to “tune,” or adjust, the controller parameters to meet the design spec
ifications. This intuitive design, as it turns out, is sufficient for many control applications.
To this day, PID control is still the predominant method in industry and is found in over 95 percent of indus
trial applications. Its success can be attributed to the simplicity, efficiency, and effectiveness of this method.
˙
e
u K e K e K e
P I D
= + +
∫
˙
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.3
The Role of Control Theory
To design a controller that makes a system behave in a desirable manner, we need a way to predict the behav
ior of the quantities of interest over time, specifically how they change in response to different inputs.
Mathematical models are most oftenly used to predict future behavior, and control system design methodolo
gies are based on such models. Understanding control theory requires engineers to be well versed in basic
mathematical concepts and skills, such as solving differential equations and using Laplace transform. The role
of control theory is to help us gain insight on how and why feedback control systems work and how to sys
tematically deal with various design and analysis issues. Specifically, the following issues are of both practi
cal importance and theoretical interest:
1.Stability and stability margins of closedloop systems.
2.How fast and smooth the error between the output and the set point is driven to zero.
3.How well the control system handles unexpected external disturbances, sensor noises, and internal dynamic
changes.
In the following, modeling and analysis are first introduced, followed by an overview of the classical design
methods for singleinput singleoutput plants, design evaluation methods, and implementation issues.
Alternative design methods are then briefly presented. Finally, For the sake of simplicity and brevity, the dis
cussion is restricted to linear, time invariant systems. Results maybe found in the literature for the cases of lin
ear, timevarying systems, and also for nonlinear systems, systems with delays, systems described by partial
differential equations, and so on; these results, however, tend to be more restricted and case dependent.
MATHEMATICAL DESCRIPTIONS
Mathematical models of physical processes are the foundations of control theory. The existing analysis and
synthesis tools are all based on certain types of mathematical descriptions of the systems to be controlled, also
called plants. Most require that the plants are linear, causal, and time invariant. Three different mathematical
models for such plants, namely, linear ordinary differential equation, state variable or state space description,
and transfer function are introduced below.
Linear Differential Equations
In control system design the most common mathematical models of the behavior of interest are, in the time
domain, linear ordinary differential equations with constant coefficients, and in the frequency or transform
domain, transfer functions obtained from time domain descriptions via Laplace transforms.
Mathematical models of dynamic processes are often derived using physical laws such as Newton’s and
Kirchhoff’s. As an example consider first a simple mechanical system, a spring/mass/damper. It consists of a
weight m on a spring with spring constant k, its motion damped by friction with coefficient f (Fig. 19.1.2).
19.4
CONTROL SYSTEMS
FIGURE 19.1.1 Feedback control configuration with cruise control as an example.
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.4
CONTROL SYSTEM DESIGN
19.5
If y(t) is the displacement from the resting position and u(t) is the force applied, it can be shown using
Newton’s law that the motion is described by the following linear, ordinary differential equation with constant
coefficients:
where with initial conditions
Note that in the next subsection the trajectory y(t) is determined, in terms of the system parameters, the initial
conditions, and the applied input force u(t), using a methodology based on Laplace transform. The Laplace
transform is briefly reviewed in Appendix A.
For a second example consider an electric RLC circuit with i(t) the input current of a current source, and
v(t) the output voltage across a load resistance R. (Fig. 19.1.3)
Using Kirchhoff’s laws one may derive:
which describes the dependence of the output voltage v(t) to the input current i(t). Given i(t) for t ≥ 0, the ini
tial values v(0) and (0) must also be given to uniquely define v(t) for t ≥ 0.
It is important to note the similarity between the two differential equations that describe the behavior of a
mechanical and an electrical system, respectively. Although the interpretation of the variables is completely differ
ent, their relations described by the same linear, secondorder differential equation with constant coefficients. This
fact is well understood and leads to the study of mechanical, thermal, fluid systems via convenient electric circuits.
State Variable Descriptions
Instead of working with many different types of higherorder differential equations that describe the behavior of
the system, it is possible to work with an equivalent set of standardized firstorder vector differential equations
that can be derived in a systematic way. To illustrate, consider the spring/mass/damper example. Let x
1
(t) = y(t),
x
2
(t) = (t) be new variables, called state variables. Then the system is equivalently described by the equations
x t x t x t
f
m
x t
k
m
x t
m
u
1 2 2 2 1
1
( ) ( ) ( ) ( ) ( )= =
−
− +and (( )t
˙
y
˙
v
˙˙
( )
˙
( ) ( ) ( )v t
R
L
v t
LC
v t
R
LC
i t+ + =
1
y t y y
dy t
dt
dy
dt
y
t t
( ) ( )
( ) ( )
( )
= =
= = = = =
0 0 0
0
0
0and
y
y
1
˙
( ) ( )y t dy t dt=
∆
/
˙˙
( )
˙
( ) ( ) ( )y t
f
m
y t
k
m
y t
m
u t+ + =
1
FIGURE 19.1.2 Spring, mass, and damper system.
FIGURE 19.1.3 RLC circuit.
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.5
with initial conditions x
1
(0) = y
0
and x
2
(0) = y
1
. Since y(t) is of interest, the output equation y(t) = x
1
(t) is also
added. These can be written as
which are of the general form
Here x(t) is a 2 × 1 vector (a column vector) with elements the two state variables x
1
(t) and x
2
(t). It is called
the state vector. The variable u(t) is the input and y(t) is the output of the system. The first equation is a vec
tor differential equation called the state equation. The second equation is an algebraic equation called the out
put equation. In the above example D = 0; D is called the direct link, as it directly connects the input to the
output, as opposed to connecting through x(t) and the dynamics of the system. The above description is the
state variable or state space description of the system. The advantage is that, system descriptions can be writ
ten in a standard form (the state space form) for which many mathematical results exist. We shall present a
number of them in this section.
A state variable description of a system can sometimes be derived directly, and not through a higherorder
differential equation. To illustrate, consider the circuit example presented above: using Kirchhoff’s current law
and from the voltage law
If the state variables are selected to be x
1
= v
c
, x
2
= i
L
, then the equations may be written as
where v = Ri
L
= Rx
2
is the output of interest. Note that the choice of state variables is not unique. In fact, if we
start from the secondorder differential equation and set and , we derive an equivalent state vari
able description,namely,
Equivalent state variable descriptions are obtained by a change in the basis (coordinate system) of the vector
state space. Any two equivalent representations
x Ax Bu y Cx Du x Ax Bu y Cx Du= + = + = + = +,,and
v
x
x
=
[ ]1 0
1
2
x
x
LC R L
x
x
1
2
1
2
0 1
1
=
− −
+
//
00
R LC
i
/
x v
2
=
˙
x v
1
=
v R
x
x
=
[ ]0
1
2
x
x
C
L R L
x
x
1
2
1
2
0 1
1
1
=
−
−
+
/
//
//C
i
0
L
di
dt
Ri v
L
L c
= − +
i C
dv
dt
i i
c
c
L
= = −
˙
( ) ( ) ( ),( ) ( ) ( )x t Ax t Bu t y t Cx t Du t= + = +
y t
x t
x t
( ) [ ]
( )
( )
=
1 0
1
2
x t
x t
k m f m
x t
x
1
2
1
2
0 1
( )
( )
//
( )
(
=
− −
tt
m
u t
)
/
( )
+
0
1
19.6
CONTROL SYSTEMS
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.6
are related by , and where P is a square and nonsingular matrix.
Note that state variables can represent physical quantities that may be measured, for instance, x
1
= v
c
voltage,
x
2
= i
L
current in the above example; or they can be mathematical quantities, which may not have direct phys
ical interpretation.
Linearization.The linear models studied here are
very useful not only because they describe linear
dynamical processes, but also because they can be
approximations of nonlinear dynamical processes in the
neighborhood of an operating point. The idea in linear
approximations of nonlinear dynamics is analogous to
using Taylor series approximations of functions to
extract a linear approximation. A simple example is that
of a simple pendulum sin x
1
, where for
small excursions from the equilibrium at zero, sin x
1
is
approximately equal to x
1
and the equations become lin
ear, namely,
Transfer Functions
The transfer function of a linear, timeinvariant system is the ratio of the Laplace transform of the output
Y(s) to the Laplace transform of the corresponding input U(s) with all initial conditions assumed to be zero
(Fig. 19.1.4).
From Differential Equations to Transfer Functions.Let the equation
with some initial conditions
describe a process of interest, for example, a spring/mass/damper system; see previous subsection.
Taking Laplace transform of both sides we obtain
where Y(s) = L{y(t)} and U(s) = L{u(t)}. Combining terms and solving with respect to Y(s) we obtain:
Assuming the initial conditions are zero,
where G(s) is the transfer function of the system defined above.
We are concerned with transfer functions G(s) that are rational functions, that is, ratios of polynomials in s,
G(s) = n(s)/d(s). We are interested in proper G(s) where G(s) < ∞. Proper G(s) have degree n(s) ≤ degree d(s).
lim
a→∞
Y s U s G s
b
s a s a
( )/( ) = =
+ +
( )
0
2
1 0
Y s
b
s a s a
U s
s a y y
s a s
( ) ( )
( ) ( ) ( )
=
+ +
+
+ +
+
0
2
1 0
1
2
1
0 0
++ a
0
[ ( ) ( ) ( )] [ ( ) ( )] ( )s y s sy y a sY s y a Y s
2
1 0
0 0 0− − + − + = b
b U s
0
( )
y t y
dy t
dt
dy
dt
y
t t
( ) ( )
( ) ( )
( )
= =
= = =
0 0
0
0
0and
d y t
dt
a
dy t
dt
a y t b u t
2
2
1 0 0
( ) ( )
( ) ( )+ + =
˙
,
˙
.x x x kx
1 2 2 1
= = −
˙
,
˙
x x x k
1 2 2
= = −
x Px=
A PAP B PB C CP D D= = = =
− −1 1
,,,
CONTROL SYSTEM DESIGN
19.7
FIGURE 19.1.4 The transfer function model.
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.7
In most cases degree n(s) < degree d(s), which case G(s) is called strictly proper. Consider the transfer
function
Note that the system described by this G(s) (Y(s) = G(s)U(s)) is described in the time domain by the following
differential equation:
y
(n)
(t) + a
n−1
y
(n−1)
(t) +
…
+ a
1
y
(1)
(t) + a
0
y(t) = b
m
u
(m)
(t) +
…
+ b
1
u
(1)
(t) + b
0
u(t)
where y
(n)
(t) denotes the nth derivative of y(t) with respect to time t. Taking Laplace transform of both sides of this
differential equation, assuming that all initial conditions are zero, one obtains the above transfer function G(s).
From State Space Descriptions to Transfer Functions.Consider
with x(0) initial conditions; x(t) is in general an ntuple, that is a (column) vector with n elements. Taking
Laplace transform of both sides of the state equation:
sX(s) – x(0) = AX(s) + BU(s) or (sI
n
– A)X(s) = BU(s) + x(0)
where I
n
is the n × n identity matrix; it has 1 on all diagonal elements and 0 everywhere else, e.g.,
.
Then
X(s) = (sI
n
− A)
−1
BU(s) + (sI
n
− A)
−1
x(0)
Taking now Laplace transform on both sides of the output equation we obtain Y(s) = CX(s) + DU(s).
Substituting we obtain,
Y(s) = [C(sI
n
− A)
−1
B + D]U(s) + C(sI − A)
−1
x(0)
The response y(t) is the inverse Laplace of Y(s). Note that the second term on the righthand side of the expres
sion depends on x(0) and it is zero when the initial conditions are zero, i.e., when x(0) = 0. The first term
describes the dependence of Y on U and it is not difficult to see that the transfer function G(s) of the systems is
G(s) = C(sI
n
− A)
−1
B + D
Example
Consider the spring/mass/damper example discussed previously with state variable description
If m = 1, f = 3, k = 2, then
and its transfer function G(s) (Y(s) = G(s)U(s)) is
as before.
G s C sI A B
s
s
( ) ( ) [ ]= − =
−
− −
−
2
1
1 0
0
0
0 1
2 3
=
−
+
−1
0
1
1 0
1
2 3
[ ]
s
s
=
+ +
+
−
−1
2
0
1
1 0
1
3 2
3 1
2
0
1
[ ]
s s
s
s
=
+ +
1
3 2
2
s s
A B C=
− −
=
=
0 1
2 3
0
1
1 0,,[ ]
˙
,.x Ax Bu y Cx= + =
I
2
1 0
0 1
=
˙
( ) ( ) ( ),( ) ( ) ( )x t Ax t Bu t y t Cx t Du t= + = +
G s
b s b s b s b
s a s a
m
m
m
m
n
n
n
( ) =
+ + + +
+ + +
−
−
−
−
1
1
1 0
1
1
1
ss a
m n
+
≤
0
with
19.8
CONTROL SYSTEMS
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.8
Using the state space description and properties of Laplace transform an explicit expression for y(t) in terms
of u(t) and x(0) may be derived. To illustrate, consider the scalar case with z(0) initial condi
tion. Using Laplace transform:
from which
Note that the second term is a convolution integral. Similarly in the vector case, given
it can be shown that
and
Notice that e
At
= L
1
{(sI − A)
−1
}. The matrix exponential e
At
is defined by the (convergent) series
Poles and Zeros.The n roots of the denominator polynomial d(s) of G(s) are the poles of G(s). The m roots
of the numerator polynomial n(s) of G(s) are (finite) zeros of G(s).
Example
(Fig. 19.1.5)
G(s) has one (finite) zero at −2 and two complex conjugate poles at −1 ± j
In general, a transfer function with m zeros and n poles can be written as
where k is the gain.
Frequency Response
The frequency response of a system is given by its transfer function G(s) evaluated at s = jw, that is, G( jw). The
frequency response is a very useful means characterizing a system, since typically it can be determined experi
mentally, and since control system specifications are frequently expressed in terms of the frequency response.
When the poles of G(s) have negative real parts, the system turns out to be boundedinput/boundedoutput
G s k
s z s z
s p s p
m
n
( )
( ) ( )
( ) ( )
=
− −
− −
1
1
G s
s
s s
s
s
s
s j s j
( )
( ) ( )( )
=
+
+ +
=
+
+ +
=
+
+ − + +
2
2 2
2
1 1
2
1 1
2 2
e I e
A t A t
k
I
t
k
A
At At
k k k
k
k
= + + + + + = +
=
∞
∑
2 2
1
2!!!
y t Ce x Ce Bu d Du t
At A t
t
( ) ( ) ( ) ( )
( )
= + +
−
∫
0
0
τ
τ τ
x t e x e Bu d
At A t
t
( ) ( ) ( )
( )
= +
−
∫
0
τ
τ τ
0
x t Ax t Bu t y t Cx t B t u t( ) ( ) ( ),( ) ( ) ( ) ( )= + = +
z t L Z s e z e bu d
at a t
t
( ) { ( )} ( ) ( )
( )
= = +
− −
∫
1
0
0
τ
τ τ
Z s
s a
z
b
s a
U s( ) ( ) ( )=
−
+
−
1
0
˙
z az bu= +
CONTROL SYSTEM DESIGN
19.9
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.9
(BIBO) stable. Under these conditions the frequency
response G( jw) has a clear physical meaning, and this
fact can be used to determine G( jw) experimentally. In
particular, it can be shown that if the input u(t) = k sin
(w
o
t) is applied to a system with a stable transfer function
G(s) (Y(s) = G(s)U(s)), then the output y(t) at steady state
(after all transients have died out) is given by
y
ss
(t) = k G(w
o
) sin [w
o
t + q(w
q
)]
where G(w
o
) denotes the magnitude of G( jw
o
) and q(w
o
) =
arg G( jw
o
) is the argument or phase of the complex quan
tity G( jw
o
). Applying sinusoidal inputs with different fre
quencies w
o
and measuring the magnitude and phase of
the output at steady state, it is possible to determine the
full frequency response of the system G( jw
o
) = G(w
o
) e
jq(w
o
)
.
ANALYSIS OF DYNAMICAL BEHAVIOR
System Response,Modes and Stability
It was shown above how the response of a system to an input and under some given initial conditions can be cal
culated from its differential equation description using Laplace transforms. Specifically, y(t) = L
−1
{Y (s)} where
with n(s)/d(s) = G(s), the system transfer function; the numerator m(s) of the second term depends on the ini
tial conditions and it is zero when all initial conditions are zero, i.e., when the system is initially at rest.
In view now of the partial fraction expansion rules, see Appendix A, Y(s) can be written as follows:
This expression shows real poles of G(s), namely, p
1
, p
2
,…,and it allows for multiple poles p
i
; it also shows
complex conjugate poles a ± jb written as secondorder terms. I(s) denotes the terms due to the input U(s);
they are fractions with poles the poles of U(s). Note that if G(s) and U(s) have common poles they are com
bined to form multiplepole terms.
Taking now the inverse Laplace transform of Y(s):
where i(t) depends on the input. Note that the terms of the form ct
k
e
p
i
t
are the modes of the system. The sys
tem behavior is the aggregate of the behaviors of the modes. Each mode depends primarily on the location of
the pole p
i
; the location of the zeros affects the size of its coefficient c.
If the input u(t) is a bounded signal, i.e., u(t) < ∞ for all t, then all the poles of I(s) have real parts that are
negative or zero, and this implies that I(t) is also bounded for all t. In that case, the response y(t) of the system
will also be bounded for any bounded u(t) if and only if all the poles of G(s) have strictly negative real parts.
Note that poles of G(s) with real parts equal to zero are not allowed, since if U(s) also has poles at the same
locations, y(t) will be unbounded. Take, for example, G(s) = 1/s and consider the bounded step input U(s) = 1/s;
the response y(t) = t, which is not bounded.
Having all the poles of G(s) located in the open left half of the splane is very desirable and it corre
sponds to the system being stable. In fact, a system is boundedinput, boundedoutput (BIBO) stable if and
only if all poles of its transfer function have negative real parts. If at least one of the poles has positive
y t L Y s c e c e te e bt bt i t
p t
i
p t p t
at
i i
( ) { ( )} ( ) [( )sin ( )cos ] ( )= = + + + ⋅ + + ⋅ + ⋅ + +
−1
1 1
1
L L L
Y s
c
s p
c
s p
c
s p
b s b
s a s a
I s
i
i
i
i
( )
( )
( )=
−
+ +
−
+
−
+ +
+
+ +
+ +
1
1
1 2
2
1 0
2
1 0
L L L
Y s
n s
d s
U s
m s
d s
( )
( )
( )
( )
( )
( )
= +
19.10
CONTROL SYSTEMS
FIGURE 19.1.5 Complex conjugate poles of G(s).
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.10
real part, then the system in unstable. If a pole has zero real part, then the term marginally stable is some
times used.
Note that in a BIBO stable system if there is no forcing input, but only initial conditions are allowed to excite
the system, then y(t) will go to zero as t goes to infinity. This is a very desirable property for a system to have,
because nonzero initial conditions always exist in most real systems. For example, disturbances such as interfer
ence may add charge to a capacitor in an electric circuit, or a sudden brief gust of wind may change the heading
of an aircraft. In a stable system the effect of the disturbances will diminish and the system will return to its pre
vious desirable operating condition. For these reasons a control system should first and foremost be guaranteed
to be stable, that is, it should always have poles with negative real parts. There are a many design methods to sta
bilize a system or if it is initially stable to preserve its stability, and several are discussed later in this section.
Response of First and Second Order Systems
Consider a system described by a firstorder differential equation, namely, and let y(0) = 0
In view of the previous subsection, the transfer function of the system is
and the response to a unit step input q(t) (q(t) = 1 for t ≥ 0, q(t) = 0 for t < 0) may be found as follows:
Note that the pole of the system is p = −a
0
(in Fig. 19.1.6 we have assumed that a
0
> 0). As that pole moves to
the left on the real axis, i.e., as a
0
becomes larger, the system becomes faster. This can be seen from the fact
that the steady state value of the system response
is approached by the trajectory of y(t) faster, as a
0
becomes
larger. To see this, note that the value 1 − e
−1
is attained at
time t = 1/a
0
, which is smaller as a
0
becomes larger. t is the
time constant of this firstorder system; see below for further
discussion of the time constant of a system.
We now derive the response of a secondorder system to a
unit step input (Fig. 19.1.7). Consider a system described by
, which gives rise to the transfer
function:
Notice that the steadystate value of the response to a unit step is
note that this normalization or scaling to 1 is in fact the reason for selecting the constant numerator to be a
0
.
G(s) above does not have any finite zeros—only poles—as we want to study first the effect of the poles on the
system behavior. We shall discuss the effect of adding a zero or an extra pole later.
y sG s
s
ss
s
= =
→
lim ( )
0
1
1
G s
a
s a s a
( ) =
+ +
0
2
1 0
˙˙
( )
˙
( ) ( ) ( )y t a y t a y t a u t+ + =
1 0 0
y y t
ss
t
= =
→∞
lim ( ) 1
y t L Y s L G s U s L
a
s a s
( ) { ( )} { ( ) ( )}= = =
+
−
− − −1 1 1
0
0
1
= +
−
+
= −
−
[L
s s a
1
0
1 1
1 e
e q t
a t−
0
] ( )
G s
a
s a
( ) =
+
0
0
˙
( ) ( ) ( )y t a y t a u t+ =
0 0
CONTROL SYSTEM DESIGN
19.11
FIGURE 19.1.6 Pole location of a firstorder
system.
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.11
It is customary, and useful as we will see, to write the above transfer function as
where z is the damping ratio of the system and w
n
is the (undamped) natural frequency of the system, i.e., the
frequency of oscillations when the damping is zero.
The poles of the system are
When z > 1 the poles are real and distinct and the unit step response approaches its steadystate value of 1
without overshoot. In this case the system is overdamped. The system is called critically damped when z = 1
in which case the poles are real, repeated, and located at −zw
n
.
The more interesting case is when the system is underdamped (z < 1). In this case the poles are complex
conjugate and are given by
The response to a unit step input in this case is
where q = cos
−1
z = tan
−1
,
and q(t) is the step function. The response to an
impulse input (u(t) = d(t)) also called the impulse response h(t) of the system is given in this case by
The secondorder system is parameterized by the two parameters z and w
n
. Different choices for z and w
n
lead
to different pole locations and to different behavior of the (modes of) the system. Fig. 19.1.8 shows the rela
tion between the parameters and the pole location.
h t
e
t q t
n
t
n
n
( ) sin=
−
−
( )
−
ω
ζ
ω ζ
ζω
1
1
2
2
( )
(/),1 1
2 2
− = −ζ ζ ω ω ζ
d n
y t
e
t q t
n
t
d
( ) ( ) ( )= −
−
+
−
1
1
2
ζω
ζ
ω θsin
p j j
n n d1 2
2
1
,
= − ± − = +ζω ω ζ σ ω
p
n n1 2
2
1
,
= − ± −ζω ω ζ
G s
s s
n
n n
( ) =
+ +
ω
ζω ω
2
2 2
2
19.12
CONTROL SYSTEMS
FIGURE 19.1.7 Step response of a firstorder plant.
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.12
Time Constant of a Mode and of a System.The time constant of a mode ce
pt
of a system is the time value
that makes pt = 1, i.e., t =1/p. For example, in the above fistorder system we have seen that t = 1/a
0
= RC.
A pair of complex conjugate poles p
1,2
= s ± jw give rise to the term of the form Ce
st
sin (wt + q). In this case,
t = 1/s, i.e., t is again the inverse of the distance of the pole from the imaginary axis. The time constant of a
system is the time constant of its dominant modes.
Transient Response Performance Specifications for a SecondOrder Underdamped System
For the system
and a unit step input, explicit formulas for important measures of performance of its transient response can be
derived. Note that the steady state is
The rise time t
r
shows how long it takes for the system’s output to rise from 0 to 66 percent of its final valu
e
(equal to 1 here) and it can be shown to be where q = cos
–1
z and w
d
= w
n
. The set
tling time t
s
is the time required for the output to settle within some percentage, typically 2 percent or 5 percent,
of its final value. is the 2 percent settling time ( is the 5 percent settling time). Before
the underdamped system settles, it will overshoot its final value. The peak time t
p
measures the time it takes
for the output to reach its first (and highest) peak value. M
p
measures the actual overshoot that occurs in
percentage terms of the final value. M
p
occurs at time t
p
, which is the time of the first and largest overshoot.
It is important to notice that the overshoot depends only on z. Typically, tolerable overshoot values are between
2.5 percent and 25 percent, which correspond to damping ratio z between 0.8 and 0.4.
t M e
p
d
p
= =
− −
π
ω
ζπ ζ
,%
/
100
1
2
t
s n
≅ 3/ζω
t
s n
≅ 4/ζω
L −ζ
2
t
r n
= −( )/,π θ ω
y sG s
s
ss
s
= =
→
lim ( )
0
1
1
G s
s s
n
n n
( ) =
+ +
ω
ζω ω
2
2 2
2
CONTROL SYSTEM DESIGN
19.13
FIGURE 19.1.8 Relation between pole location and parameters.
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.13
Effect of Additional Poles and Zeros
The addition of an extra pole in the lefthalf splane (LHP) tends to slow the system down—the rise time of
the system, for example, will become larger. When the pole is far to the left of the imaginary axis, its effect
tends to be small. The effect becomes more pronounced as the pole moves toward the imaginary axis.
The addition of a zero in the LHP has the opposite effect, as it tends to speed the system up. Again the effect
of a zero far away to the left of the imaginary axis tends to be small. It becomes more pronounced as the zero
moves closer to the imaginary axis.
The addition of a zero in the righthalf splane (RHP) has a delaying effect much more severe than the
addition of a LHP pole. In fact a RHP zero causes the response (say, to a step input) to start toward the wrong
direction. It will move down first and become negative, for example, before it becomes positive again and
starts toward its steadystate value. Systems with RHP zeros are called nonminimum phase systems (for rea
sons that will become clearer after the discussion of the frequency design methods) and are typically difficult
to control. Systems with only LHP poles (stable) and LHP zeros are called minimum phase systems.
CLASSICAL CONTROL DESIGN METHODS
In this section, we focus on the problem of controlling a singleinput and singleoutput (SISO) LTI plant. It is
understood from the above sections that such a plant can be represented by a transfer function G
p
(s). The
closedloop system is shown in Fig. 19.1.9.
The goal of feedback control is to make the output of the plant, y, follow the reference input r as closely as
possible. Classical design methods are those used to determine the controller transfer function G
c
(s) so that the
closedloop system, represented by the transfer function:
has desired characteristics.
Design Specifications and Constraints
The design specifications are typically described in terms of step response, i.e., r is the set point described as
a steplike function. These specifications are given in terms of transient response and steadystate error, assum
ing the feedback control system is stable. The transient response is characterized by the rise time, i.e., the time
it takes for the output to reach 66 percent of its final value, the settling time, i.e., the time it takes for the out
put to settle within 2 percent of its final value, and the percent overshoot, which is how much the output
exceeds the setpoint r percentagewise during the period that y converges to r. The steadystate error refers to
the difference, if any, between y and r as y reaches its steadystate value.
There are many constraints a control designer has to deal with in practice, as shown in Fig. 19.1.10. They
can be described as follows:
1.Actuator Saturation: The input u to the plant is physically limited to a certain range, beyond which it “sat
urates,” i.e., becomes a constant.
2.Disturbance Rejection and Sensor Noise Reduction: There are always disturbances and sensor noises in the
plant to be dealt with.
G s
G s G s
G s G s
CL
c p
c p
( )
( ) ( )
( ) ( )
=
+1
19.14
CONTROL SYSTEMS
FIGURE 19.1.9 Feedback control configuration.
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.14
3.Dynamic Changes in the Plant: Physical systems are almost never truly linear nor time invariant.
4.Transient Profile: In practice, it is often not enough to just move y from one operating point to another. How
it gets there is sometimes just as important. Transient profile is a mechanism to define the desired trajectory
of y in transition, which is of great practical concerns. The smoothness of y and its derivatives, the energy
consumed, the maximum value and the rate of change required of the control action are all influenced by the
choice of transient profile.
5.Digital Control: Most controllers are implemented today in digital forms, which makes the sampling rate
and quantization errors limiting factors in the controller performance.
Control Design Strategy Overview
The control strategies are summarized here in ascending order of complexity and, hopefully, performance.
1.OpenLoop Control:If the plant transfer function is
known and there is very little disturbance, a simple
open loop controller, as shown in Fig. 19.1.11,
would satisfy most design requirements.
Where G
c
(s) is an approximate inverse of G
p
(s).
Such control strategy has been used as an economic
means in controlling stepper motors, for example.
2.Feedback Control with a Constant Gain:With sig
nificant disturbance and dynamic variations in the
plant, feedback control, as shown in Fig. 19.1.9 is the only choice; see also Appendix B. Its simplest
form is G
c
(s) = k, or u = ke, where k is a constant. Such proportional controller is very appealing
because of its simplicity. The common problems with this controller are significant steadystate error
and overshoot.
3.ProportionalIntegralDerivative Controller:To correct the above problems with the constant gain con
troller, two additional terms are added:
This is the wellknown PID controller, which is used by most engineers in industry today. The design can
be quite intuitive: the proportional term usually plays the key role, with the integral term added to reduce/elim
inate the steadystate error and the derivative term the overshoot. The primary drawbacks of PID is that the
integrator introduces phase lag that could lead to stability problems and the differentiator makes the controller
sensitive to noise.
u k e k e k e G s k k s k s
p i d c p i d
= + + = + +
∫
or ( )/
CONTROL SYSTEM DESIGN
19.15
FIGURE 19.1.10 Closedloop simulator setup.
FIGURE 19.1.11 Openloop control configuration.
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.15
4.Root Locus Method: A significant portion of most current control textbooks is devoted to the question of how
to place the poles of the closedloop system in Fig.19.1.9 at desired locations, assuming we know where they
are. Root locus is a graphical technique to manipulate the closedloop poles given the openloop transfer func
tion. This technique is most effective if disturbance rejection, plant dynamical variations, and sensor noise are
not to be considered. This is because these properties cannot be easily linked to closed loop pole locations.
5.LoopShaping Method: Loopshaping [5] refers to the manipulation of the loop gain frequency response,
L(jw) = G
p
(jw)G
c
(jw), as a control design tool. It is the only existing design method that can bring most of
design specifications and constraints, as discussed above, under one umbrella and systematically find a
solution. This makes it a very useful tool in understanding, diagnosing, and solving practical control prob
lems. The loopshaping process consists of two steps:
a.Convert all design specifications to loop gain constraints, as shown in Fig.19.1.12.
b.Find a controller G
c
(s) to meet the specifications.
Loopshaping as a concept and a design tool helped the practicing engineers greatly in improving the PID
loop performance and stability margins. For example, a PID implemented as a leadlag compensator is com
monly seen in industry today. This is where the classical control theory provides the mathematical and design
insights on why and how feedback control works. It has also laid the foundation for modern control theory.
Example
Consider a motion control system as shown
in Fig. 19.1.13 below. It consists of a digital controller,
a dc motor drive (motor and power amplifier), and a
load of 235 lb that is to be moved linearly by 12 in. in
0.3 s with an accuracy of 1 percent or better. A belt and
pulley mechanism is used to convert the motor rotation
a linear motion. Here a servo motor is used to drive the
load to perform a linear motion. The motor is coupled
with the load through a pulley.
The design process involves:
1.Selection of components including motor, power
amplifier, the beltandpulley, the feedback devices
(position sensor and/or speed sensor)
2.Modeling of the plant
3.Control design and simulation
4.Implementation and tuning
19.16
CONTROL SYSTEMS
FIGURE 19.1.12 Loopshaping.
motor
load
pulley
FIGURE 19.1.13 A Digital servo control design example.
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.16
The first step results in a system with the following parameters:
1.Electrical:
•
Winding resistance and inductance: R
a
= 0.4 mho L
a
= 8 mH (the transfer function of armature voltage to
current is (1/R
a
)/[(L
a
/R
a
)s + 1]
•
back emf constant: K
E
= 1.49 V/(rad/s)
•
power amplifier gain: K
pa
= 80
•
current feedback gain: K
cf
= 0.075 V/A
2.Mechanical:
•
Torque constant: K
t
= 13.2 inlb/A
•
Motor inertia J
m
= .05 lbin.s
2
•
Pulley radius R
p
= 1.25 in.
•
Load weight: W = 235 lb (including the assembly)
•
Total inertia J
t
= J
m
+ J
l
=0.05 + (W/g)R
p
2
= 1.0 lbin.s
2
With the maximum armature current set at 100 A, the maximum Torque = K
t
I
a,max
= 13.2 × 100 = 1320 in.lb;
the maximum angular acceleration = 1320/J
t
= 1320 rad/s
2
, and the maximum linear acceleration = 1320 × R
p
=
1650 in./s
2
= 4.27 g’s (1650/386). As it turned out, they are sufficient for this application.
The second step produces a simulation model (Fig. 19.1.14).
A simplified transfer function of the plant, from the control input, v
c
(in volts), to the linear position output,
x
out
(in inches), is
An open loop controller is not suitable here because it cannot handle the torque disturbances and the
inertia change in the load. Now consider the feedback control scheme in Fig. 19.1.9 with a constant con
troller, u = ke. The root locus plot in Fig. 19.1.15 indicates that, even at a high gain, the real part of the closed
loop poles does not exceed −1.5, which corresponds to a settling time of about 2.7 s. This is far slower than
desired.
In order to make the system respond faster, the closedloop poles must be moved further away from the jw
axis. In particular, a settling time of 0.3 s or less corresponds to the closedloop poles with real parts smaller
than −13.3. This is achieved by using a PD controller of the form
G
c
(s) = K(s + 3);K ≥ 13.3/206
will result in a settling time of less than 0.3 s.
The above PD design is a simple solution in servo design that is commonly used. There are several issues,
however, that cannot be completely resolved in this framework:
1.Lowfrequency torque disturbance induces steadystate error that affects the accuracy.
2.The presence of a resonant mode within or close to the bandwidth of the servo loop may create undesirable
vibrations.
3.Sensor noise may cause the control signal to be very noisy.
4.The change in the dynamics of the plant, for example, the inertia of the load, may require frequency tweak
ing of the controller parameters.
5.The steplike set point change results in an initial surge in the control signal and could shorten the life span
of the motor and other mechanical parts.
These are problems that most control textbook do not adequately address, but they are of significant impor
tance in practice. The first three problems can be tackled using the loopshaping design technique introduced
above. The tuning problem is an industrywide design issue and the focus of various research and development
efforts. The last problem is addressed by employing a smooth transient as the set point, instead of a steplike
setpoint. This is known as the “motion profile” is industry.
G s
s s
p
( )
( )
=
+
206
3
CONTROL SYSTEM DESIGN
19.17
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.17
1
Vc
control
signal
+/−8V
+
+
+
+
−
−
80
+/−160V
armature
voltage
2.5
.02s + 1
la
la
Armature Dynamics
13.2
1.0
total
torque
torque
disturbance
2
2
.075
Kcf
Kt
Ke
angular
accelation
angular
velocity
linear
velocity
linear
position
1
1
s
1
s
1.25
3
Vout
Xout
vel to posRpacc to vel1/Jt
back emfect
1.49
Power amp
Kpa
FIGURE 19.1.14 Simulation model of the motion control system.
19.18
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.18
Evaluation of Control Systems
Analysis of control system provides crucial insights to control practitioners on why and how feedback control
works. Although the use of PID precedes the birth of classical control theory of the 1950s by at least two
decades, it is the latter that established the control engineering discipline. The core of classical control theory
are the frequencyresponsebased analysis techniques, namely, Bode and Nyquist plots, stability margins, and
so forth.
In particular, by examining the loop gain frequency response of the system in Fig. 19.1.9, that is, L( jw) =
G
c
( jw)G
p
( jw), and the sensitivity function 1/[1 + L(jw)], one can determine the following:
1.How fast the control system responds to the command or disturbance input (i.e., the bandwidth).
2.Whether the closedloop system is stable (Nyquist Stability Theorem); If it is stable, how much
dynamic variation it takes to make the system unstable (in terms of the gain and phase change in the
plant). It leads to the definition of gain and phase margins. More broadly, it defines how robust the
control system is.
3.How sensitive the performance (or closedloop transfer function) is to the changes in the parameters of the
plant transfer function (described by the sensitivity function).
4.The frequency range and the amount of attenuation for the input and output disturbances shown in Fig. 19.1.10
(again described by the sensitivity function).
Evidently, these characteristics obtained via frequencyresponse analysis are invaluable to control engi
neers. The efforts to improve these characteristics led to the leadlag compensator design and, eventually, loop
shaping technique described above.
Example:
The PD controller in Fig. 19.1.10 is known to the sensitive to sensor noises. A practical cure to
this problem is: add a low pass filter to the controller to attenuate highfrequency noises, that is,
G s
s
s
c
( )
.( )
=
+
+
13 3 3
206
133
1
2
CONTROL SYSTEM DESIGN
19.19
FIGURE 19.1.15 Root locus plot of the servo design problem.
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.19
The loop gain transfer function is now
The bandwidth of the low pass filter is chosen to be one decade higher than the loop gain bandwidth to main
tain proper gain and phase margins. The Bode plot of the new loop gain, as shown in Fig. 19.1.16, indicates that
(a) the feedback system has a bandwidth 13.2 rad/s, which corresponds to a 0.3 s settling time as specified and
(b) this design has adequate stability margins (gain margin is 26 dB and phase margin is 79°).
Digital Implementation
Once the controller is designed and simulated successfully, the next step is to digitize it so that it can be pro
grammed into the processor in the digital control hardware. To do this:
1.Determine the sampling period T
s
and the number of bits used in analogtodigital converter (ADC) and
digitaltoanalog converter (DAC).
2.Convert the continuous time transfer function G
c
(s) to its corresponding form in discrete time transfer func
tion G
cd
(z) using, for example, the Tustin’s method, s = (1/T)(z − 1)/(z + 1).
3.From G
cd
(z), derive the difference equation, u(k) = g(u(k − 1), u(k − 2), . . . y(k), y(k – 1), . . .), where g is
a linear algebraic function.
After the conversion, the sampled data system, with the plant running in continuous time and the controller
in discrete time, should be verified in simulation first before the actual implementation. The quantization error
and sensor noise should also be included to make it realistic.
The minimum sampling frequency required for a given control system design has not been established ana
lytically. The rule of thumb given in control textbooks is that f
s
= 1/T
s
should be chosen approximately 30 to
60 times the bandwidth of the closedloop system. Lowersampling frequency is possible after careful tuning
but the aliasing, or signal distortion, will occur when the data to be sampled have significant energy above the
L s G s G s
s
s
p c
( ) ( )= =
+
( )
.13 3
133
1
2
19.20
CONTROL SYSTEMS
FIGURE 19.1.16 Bode plot evaluation of the control design.
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.20
Nyquist frequency. For this reason, an antialiasing filter is often placed in front of the ADC to filter out the
highfrequency contents in the signal.
Typical ADC and DAC chips have 8, 12, and 16 bits of resolution. It is the length of the binary number used
to approximate an analog one. The selection of the resolution depends on the noise level in the sensor signal and
the accuracy specification. For example, the sensor noise level, say 0.1 percent, must be below the accuracy spec
ification, say 0.5 percent. Allowing one bit for the sign, an 8bit ADC with a resolution of 1/2
7
, or 0.8 percent, is
not good enough; similarly, a 16bit ADC with a resolution. 0.003 percent is unnecessary because several bits
are “lost” in the sensor noise. Therefore, a 12bit ADC, which has a resolution of 0.04 percent, is appropriate
for this case. This is an example of “error budget,” as it is known among designers, where components are
selected economically so that the sources of inaccuracies are distributed evenly.
Converting G
c
(s) to G
cd
(z) is a matter of numerical integration. There have been many methods suggested,
some are too simple and inaccurate (such as the Euler’s forward and backward methods), others are too com
plex. The Tustin’s method suggested above, also known as trapezoidal method or bilinear transformation, is a
good compromise. Once the discrete transfer function G
cd
(z) is obtained, finding the corresponding difference
equation that can be easily programmed in C is straightforward. For example, given a controller with input e(k)
and output u(k), and the transfer function
the corresponding inputoutput relationship is
or equivalently, (1 + z
−1
)u(k) = (1 + 2z
−1
)e(k). That is, u(k) = −u(k − 1) + e(k) + 2e(k − 1).
Finally, the presence of the sensor noise usually requires that an antialiasing filter be used in front of the
ADC to avoid distortion of the signal in ADC. The phase lag from such a filter must not occur at the crossover
frequency (bandwidth) or it will reduce the stability margin or even destabilize the system. This puts yet another
constraint on the controller design.
ALTERNATIVE DESIGN METHODS
Nonlinear PID
Using nonlinear PID (NPID) is an alternative to PID for better performance. It maintains the simplicity and intu
ition of PID, but empowers it with nonlinear gains. An example of NPID is shown in Fig. 19.1.17. The need for
the integral control is reduced, by making the proportional gain larger, when the error is small. The limited
u k
z
z
e k( ) ( )=
+
+
−
−
1 2
1
1
1
G z
z
z
z
z
cd
( ) =
+
+
=
+
+
−
−
2
1
1 2
1
1
1
CONTROL SYSTEM DESIGN
19.21
FIGURE 19.1.17 Nonlinear PID for a power converter control problem.
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.21
authority integral control has its gain zeroed outside a small interval around the origin to reduce the phase lag.
Finally the differential gain is reduced for small errors to reduce sensitivities to sensor noise. See Ref. 8.
State Feedback and ObserverBased Design
If the state space model of the plant
is available, the poleplacement design can be achieved via state feedback
u = r + Kx
where K is the gain vector to be determined so that the eigenvalues of the closedloops system
are at the desired locations, assuming they are known. Usually the state vector is not available through mea
surements and the state observer is of the form
where is the estimate of x and L is the observer gain vector to be determined.
The state feedback design approach has the same drawbacks as those of Root Locus approach, but the use
of the state observer does provide a means to extract the information about the plant that is otherwise unavail
able in the previous control schemes, which are based on the inputoutput descriptions of the plant. This proves
to be valuable in many applications. In addition, the state space methodologies are also applicable to systems
with many inputs and outputs.
Controllability and Observability.Controllability and observability are useful system properties and are
defined as follows. Consider an nth order system described by
where A is an n × n matrix. The system is controllable if it is possible to transfer the state to any other state in
finite time. This property is important as it measures, for example, the ability of a satellite system to reorient
itself to face another part of the earth’s surface using the available thrusters; or to shift the temperature in an
industrial oven to a specified temperature. Two equivalent tests for controllability are:
The system (or the pair (A, B)) is controllable if and only if the controllability matrix C = [B, AB,…, A
n−1
B]
has full (row) rank n. Equivalently if and only if [s
i
I − A, B] has full (row) rank n for all eigenvalues s
i
of A.
The system is observable if by observing the output and the input over a finite period of time it is possible
to deduce the value of the state vector of the system. If, for example, a circuit is observable it may be pos
sible to determine all the voltages across the capacitors and all currents through the inductances by observ
ing the input and output voltages.
The system (or the pair (A, C)) is observable if and only if the observability matrix
θ=
−
C
CA
CA
n 1
x Ax Bu z Mx= + =,
ˆ
x
ˆ
˙
ˆ
(
ˆ
)
ˆ ˆ
x Ax Bu L y y
y Cx Du
= + + −
= +
˙
( )x A BK x Br
y Cx Du
= + +
= +
˙
x Ax Bu
y Cx Du
= +
= +
19.22
CONTROL SYSTEMS
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.22
has full (column) rank n. Equivalently if and only if
has full (column) rank n for all eigenvalues s
i
of A.
Consider now the transfer function
G(s) = C(sI − A)
−1
B + D
Note that, by definition, in a transfer function all possible cancellations between numerator and denominator
polynomials are assumed to have already taken place. In general, therefore, the poles of G(s) are some (or all)
of the eigenvalues of A. It can be shown that when the system is both controllable and observable no cancel
lations take place and so in this case the poles of G(s) are exactly the eigenvalues of A.
Eigenvalue Assignment Design.Consider the equations: = Ax + Bu, y = Cx + Du, and u = p + kx. When
the system is controllable, K can be selected to assign the closedloop eigenvalues to any desired locations
(real or complex conjugate) and thus significantly modify the behavior of the openloop system. Many algo
rithms exist to determine such K. In the case of a single input, there is a convenient formula called
Ackermann’s formula
K = −[0,…, 0, 1] C
−1
a
d
(A)
where C = [B, . . . , A
n−1
B] is the n × n controllability matrix and the roots of a
d
(s) are the desired closedloop
eigenvalues.
Example
Let
and the desired eigenvalues be –1 ± j
Here
Note that A has eigenvalues at 0 and 5/2. We wish to determine K so that the eigenvalues of A + BK are
at −1 ± j, which are the roots of a
d
(s) = s
2
+ 2s + 2.
Here
and
Then
K C A
d
= − = − −
−
[ ] ( ) [//]0 1 1 6 13 3
1
a
a
d
A A A I( )
////
/
= + + =
+
+
=
2
2
2 2
1 2 1
1 2
2
1 2 1
1 2
2
1 0
0 1
17 4 9 2
9 2 11
C = =
[,]
/
B AB
1 3 2
1 3
C = =
[,]
/
B AB
1 3 2
1 3
A B=
=
12 1
1 2
1
1
/
,
˙
x
s I A
C
i
−
CONTROL SYSTEM DESIGN
19.23
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.23
Here
which has the desired eigenvalues.
Linear Quadratic Regulator (LQR) Problem.Consider
We wish to determine u(t), t ≥ 0, which minimizes the quadratic cost
for any initial state x(0). The weighting matrices Qand R are real, symmetric (Q= Q
T
, R = R
T
), Qand R are posi
tive definite (R > 0, Q > 0) and M
T
QMis positive semidefinite (M
T
QM≥ 0). Since R > 0, the term u
T
Ru is always
positive for any u ≠ 0, by definition. Minimizing its integral forces u(t) to remain small. M
T
QM ≥ 0 implies that
x
T
M
T
QMx is positive, but it can also be zero for some x ≠ 0, this allows some of the states to be treated as “do not
care states.” Minimizing the integral of x
T
M
T
QMx forces the states to become smaller as time progresses. It is con
venient to take Q (and R in the multiinput case) to be diagonal with positive entries on the diagonal. The above
performance index is designed so that the minimizing control input drives the states to the zero state, or as close as
possible, without using excessive control action, in fact minimizing the control energy. When (A, B, Q
1/2
M) is con
trollable and observable, the solution u
*
(t) of this optimal control problem is a state feedback control law, namely,
u
*
(t) = K
*
x(t) = −R
−1
B
T
P
c
*
x(t)
where P
c
*
is the unique symmetric positive definite solution of the algebraic Riccati equation:
A
T
P
c
+ P
c
A − P
c
BR
−1
B
T
P
c
+ M
T
QM= 0
Example.
Consider
And let
Here
Solving the Riccati equation we obtain
and
u t K x t x t x t
* *
( ) ( ) [ ] ( ),( )= = −
= −
[ ]
1
4
0 1
2 2
2 2 2
1
2
1 2
P
c
*
=
2 2
2 2 2
M C Q M QM C C R
T T
= = = =
=
,,[ ],1
1
0
1 0
1 0
0 0
== 4
J y t u t dt= +
( )
∞
∫
2 2
0
4( ) ( )
˙
,[ ]x x u y x=
+
=
0 1
0 0
0
1
1 0
J u x t M QM x u t Ru t dt
T T T
( ) ( )( ) ( ) ( )= +
∞
∫
0
x Ax Bu z Mx= + =,
A BK+ =
−
−
13 10 3
5 6 7 3
//
//
19.24
CONTROL SYSTEMS
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.24
Linear State Observers.Since the states of a system contain a great deal of useful information, knowledge
of the state vector is desirable. Frequently, however, it may be either impossible or impractical to obtain mea
surements of all states. Therefore, it is important to be able to estimate the states from available measurements,
namely, of inputs and outputs.
Let the system be
An asymptotic state estimator of the full state, also called Luenberger observer, is given by
where L is selected so that all eigenvalues of A − LC are in the LHP (have negative real parts). Note that a L
that arbitrarily assigns the eigenvalues of A − LC exists if and only if the system is observable. The observer
may be written as
which clearly shows the role of u and y;they are the inputs to the observer. If the error then
e(t) = e
[(A−LC)t]
e(0), which shows that
To determine appropriate L, note that which is the problem
addressed above in the state feedback case. One could also use the following observable version of
Ackermann’s formula, namely,
L = a
d
(A) q
–1
[0,…0,1]
T
where
The gain L in the above estimator may be determined so that it is optimal in an appropriate sense. In the fol
lowing, some of the key equations of such an optimal estimator (Linear Quadratic Gaussian (LQG)), also
known as the KalmanBucy filter, are briefly outlined.
Consider
where w and v represent process and measurement noise terms. Both w and v are assumed to be white, zero
mean Gaussian stochastic processes, i.e., they are uncorrelated in time and have expected values E[w] = 0 and
E[v] = 0. Let denote the covariances where W, V are real, symmetric and positive
definite matrices. Assume also that the noise processes w and v are independent, i.e., E[wv
T
] = 0. Also assume
that the initial state x(0) is a Gaussian random variable of known mean, E[x(0)] = x
0
, and known covariance
E[(x(0) − x
0
)(x(0) − x
0
)
T
] = P
e0
. Assume also that x(0) is independent of w and v.
Consider now the estimator
and let (A, Γ W
1/2
, C) be controllable and observable. Then the error covariance is minimized
when the filter gain L
*
= P
*
e
C
T
V
–1
, where P
*
e
denotes the symmetric, positive definite solution of the (dual to
control) algebraic Riccati equation
P
e
A
T
+ AP
e
− P
e
C
T
V
−1
CP
e
+ ΓWΓ
T
= 0
E x x x x
T
[(
ˆ
)(
ˆ
) ]− −
ˆ
˙
( )
ˆ
x A LC x Bu Ly= − + +
E ww W E vv V
T T
[ ],[ ]= =
x Ax Bu w y Cx v= + + = +Γ,
θ =
−
C
CA
CA
n 1
( ) ( ),A LC A C L A BK
T T T
− = + − = +
e t x t x t t( )
ˆ
( ) ( )→ → →∞0 or as
e t x t x t( ) ( )
ˆ
( )= −
ˆ
( )
ˆ
[,]x A LC x B LD K
u
y
= − + −
ˆ
˙
ˆ
(
ˆ
)x Ax Bu L y y= + + −
x Ax Bu y Cx Du= + = +,
CONTROL SYSTEM DESIGN
19.25
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.25
The above Riccati is the dual to the Riccati equation for optimal control and can be obtained from the optimal
control equation by making use of the substitutions:
A →A
T
, B →C
T
, M→Γ
T
,R → V,Q →W
In the state feedback control law u = Kx + r, when state measurements are not available, it is common to use
the estimate of state from a Luenberger observer. That is, given
the control law is , where is the state estimate from the observer
The closedloop system is then of order 2n since the plant and the observer are each of order n. It can be shown
that in this case, of linear output feedback control design, the design of the control law and of the gain K(using,
for example, LQR) can be carried out independent of the design of the estimator and the filter gain L (using,
for example, LQG). This is known as the separation property.
It is remarkable to notice that the overall transfer function of the compensated system that includes the state
feedback and the observer is
T(s) = (C + DK)[sI − (A + BK)]
−1
B + D
which is exactly the transfer function one would obtain if the state x were measured directly and the state
observer were not present. This is of course assuming zero initial conditions (to obtain the transfer function);
if nonzero initial conditions are present, then there is some deterioration of performance owing to observer
dynamics, and the fact that at least initially the state estimate typically contains significant error.
ADVANCED ANALYSIS AND DESIGN TECHNIQUES
This section covered some fundamental analysis and design methods in classical control theory, the
development of which was primarily driven by engineering practice and needs. Over the last few decades,
vast efforts in control research have led to the creation of modern mathematical control theory, or
advanced control, or control science. This development started with optimal control theory in the 50s and
60s to study the optimality of control design; a brief glimpse of optimal control was given above. In opti
mal control theory, a cost function is to be minimized, and analytical or computational methods are used
to derive optimal controllers. Examples include minimum fuel problem, timeoptimal control (Bang
Bang) problem, LQ, H
2
, and H
∞
, each corresponding to a different cost function. Other major branches
in modern control theory include multiinput multioutput (MIMO) control systems methodologies,
which attempt to extend wellknown SISO design methods and concepts to MIMO problems; adaptive
control, designed to extend the operating range of a controller by adjusting automatically the controller
parameters based on the estimated dynamic changes in the plants; analysis and design of nonlinear con
trol systems, and so forth.
A key problem is the robustness of the control system. The analysis and design methods in control theory
are all based on the mathematical model of the plants, which is an approximate description of physical process
es. Whether a control system can tolerate the uncertainties in the dynamics of the plants, or how much uncer
tainty it takes to make a system unstable, is studied in robust control theory, where H
2
, H
∞
, and other analysis
and design methods were originated. Even with recent progress, open problems remain when dealing with real
world applications. Some recent approaches such as in Ref. 8 attempt to address some of these difficulties in a
realistic way.
ˆ
( )
ˆ
[,]x A LC x B KD K
u
y
= − + −
ˆ
x
u Kx r= +
ˆ
x Ax Bu y Cx Du= + = +,
ˆ
x
19.26
CONTROL SYSTEMS
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.26
The corresponding description in the time domain using differential equations is (t) – (1 + e)y(t) = u(t). Solving,
using Laplace transform, we obtain sY(s) − y(0) − (1 + e)Y(s) = U(s) from which
Consider now the controller with transfer function
G s
s
s
c
( ) =
−
+
1
2
Y s
y
s s
U s( )
( )
( ) ( )
( )=
− +
+
− +
0
1
1
1e e
˙
y
APPENDIX:OPEN AND CLOSED LOOP STABILIZATION
It is impossible to stabilize an unstable system using openloop control, owing to system uncertainties. In gen
eral, closed loop or feedback control is necessary to control a system—stabilize if unstable and improve
performance—because of uncertainties that are always present. Feedback provides current information about
the system and so the controller does not have to rely solely on incomplete system information contained in a
nominal plant model. These uncertainties are system parameter uncertainties and also uncertainties induced on
the system by its environment, including uncertainties in the initial condition of the system, and uncertainties
because of disturbances and noise.
Consider the plant with transfer function
where the pole location at 1 is inaccurately known.
G s
s
( )
( )
=
− +
1
1 e
CONTROL SYSTEM DESIGN
19.27
The corresponding description in the time domain using differential equations is (t) + 2u(t) = (t) − r(t).
Solving, using Laplace transform, we obtain sU(s) − u(0) + 2U(s) = sR(s) − R(s) from which
Connect now the plant and the controller in series (openloop control)
U s
u
s
s
s
R s( )
( )
( )=
+
+
−
−
0
2
1
2
˙
r
˙
u
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.27
The overall transfer function is
Including the initial conditions
It is now clear that openloop control cannot be used to stabilize the plant:
1.First, because of the uncertainties in the plant parameters. Note that the plant pole is not exactly at +1 but
at 1 + e and so the controller zero cannot cancel the plant pole exactly.
2.Second, even if we had knowledge of the exact pole location, that is, e = 0, and
still we cannot stabilize the system because of the uncertainty in the initial conditions. We cannot, for exam
ple, select r(0) so as to cancel the unstable pole at +1 because y(0) may not be known exactly.
We shall now stabilize the above plant using a simple feedback controller.
Y s
s y r
s s s
R s( )
( ) ( ) ( )
( )( )
( )=
+ +
− +
+
+
2 0 0
1 2
1
2
Y s
s y u
s s
s
s s
R s( )
( ) ( ) ( )
[ ( )]( ) [ ( )]( )
( )=
+ +
− + +
+
−
− + +
2 0 0
1 2
1
1 2e e
T GG
s
s s
c
= =
−
− + +
1
1 2[ ( )]( )ε
19.28
CONTROL SYSTEMS
Consider a unity feedback control system with the controller being just a gain k to be determined. The closed
loop transfer function is
Working in the time domain,
– (1 + e)y = u = k(r − y)
from which
+ [k − (1 + e)]
y
= kr
Using Laplace transform we obtain
sY(s) − y(0) + [k − (1 + e)]Y(s) = kR(s)
and
Y s
y
s k
k
s k
R s( )
( )
( ) ( )
( )=
+ − +
+
+ − +
0
1 1e e
˙
y
˙
y
T s
kG s
kG s
k
s k
( )
( )
( ) ( )
=
+
=
− + +1 1 e
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.28
It is now clear that if the controller gain is selected so that k > 1 + e, then the system in stable. Is fact, we could
have worked with the nominal system to derive k > 1 for stability. For stability robustness, we take k somewhat
larger than 1 to have some safety margin and satisfy k > 1 + e for some unknown small e.
REFERENCES
1.Dorf, R. C., and R. H. Bishop, “Modern Control Systems,” 9th ed., Prentice Hall, 2001.
2.Franklin, G. F., J. D. Powell, and A. EmamiNaeimi, “Feedback Control of Dynamic Systems,” 3rd ed., AddisonWesley,
1994.
3.Kuo, B. C., “Automatic Control Systems,” 7th ed., Prentice Hall, 1995.
4.Ogata, K., “Modern Control Engineering,” 3rd ed., Prentice Hall, 1997.
5.Rohrs, C. E., J. L. Melsa, and D. G. Schultz, “Linear Control Systems,” McGrawHill, 1993.
6.Antsaklis, P. J., and A. N. Michel, “Linear Systems,” McGrawHill, 1997.
7.Goodwin, G. C., S. F. Graebe, and M. E. Salgado, “Control System Design,” Prentice Hall, 2001.
8.Gao, Z., Y. Huang, and J. Han, “An Alternative Paradigm for Control System Design,” Presented at the 40th IEEE
Conference on Decision and Control, Dec 4–7, 2001, Orlando, FL.
CONTROL SYSTEM DESIGN
19.29
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.29
ChristiansenSec.19.qxd 06:08:2004 6:43 PM Page 19.30
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